If you're willing to buy a uni-tasker, this will do the trick beautifully.

Speaking as a former stained-glass enthusiast, the cutting is easy but the smoothing to make it comfortable to drink from is what's going to be difficult.
For glass cutting, you need a tool like this. Get one with oil in it, not this cheapo kind. Basically you score the surface all around, tap with some heavy pliers to get the score in a little deeper, and then use the pliers just outside of your score line (with your other hand right inside the score) to gently break the piece off.

Now you have a sharp rim. You can lightly file the edge (in one direction, moving smoothly around the circle) with a metal nail file, which will make it safe to drink from. But it won't give you a smooth rounded lip like those glasses have. Those glasses were melted with a high temp (my guess is acetylene) torch until they rolled over and melted just a bit. You can't really do that at home unless you have some serious equipment. My butane jeweler's torch wouldn't even come close.

So, you can get the shape but not the comfort. It's probably worth trying one and seeing if you can drink from it comfortably without the melted edge. My guess is no.

Wow, thanks annaramma.

That seems like a pretty nice product. It looks like that glass cutter could be bolted to a workbench, which might be pretty convenient.

So the kit is just a glass cutter and polishing cloth, pretty much? I mean, I've already got a torch.

On preview: mine's propane, but I'm guessing that wouldn't work?

Propane torch would be worth a try!
I think the appeal of that kit is that it's basically a jig so you don't have to measure or use a ruler to cut in a straight line all around the bottle. But yeah, you could jerry-rig one yourself just as easily.

This is a neat idea, box. You could make some really cool glass like this. If you do it, report back with some pics!

Yeah, the kits are relatively inexpensive and make it easy to get started on this kind of project. You don't have to have much experience or equipment.

Since you already have a torch, you might want to come up with a DIY set up that would make the cutting easier. Then you could see if the torch would give you the smooth edge. I've only used a dremel tool to file down the edge (similar to the process outlined here), but a flamed lip would look much more professional.

glassES

Seconding post pictures or results!! It would so cool to be able to do this.

Yeah, let me know how that goes. I've got glass tools here and would be willing to try something like that for bar glasses.

If the propane torch doesn't do it, you might try MAPP gas; it burns hotter. It is available in yellow cylinders much like the blue propane ones.

Ooh, now there's some pressure to try to do this sooner rather than later.

That Etsy person says that their glasses are flame-cut and polished, and I get the impression they've got access to an actual glass studio. I'm kinda wondering what kind of results I can get in the garage.

Do it! You know you want too . . . all the cool kids are doing it . . .
(Is that enough peer pressure for you?)

Definitely post pics, I'm curious to see how they turn out.